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A CERTAIN P-ADIC SPECTRAL THEOREM RL Baker Department Of International Journal of Pure and Applied Mathematics Volume 75 No. 1 2012, 19-68 AP ISSN: 1311-8080 (printed version) url: http://www.ijpam.eu ijpam.eu A CERTAIN p-ADIC SPECTRAL THEOREM R.L. Baker Department of Mathematics University of Iowa Iowa City, Iowa, 52242, USA Abstract: We extend a p-adic spectral theorem of M. M. Vishik to a certain class of p-adic Banach algebras. This class includes inductive limits of finite- dimensional p-adic Banach algebras of the form B(X ), where X is a p-adic Banach space of the form X ≃ Ωp(J), J being a finite nonempty set. In particular, we present a p-adic spectral theorem for p-adic UHF algebras and p-adic TUHF algebras (Triangular UHF Algebras). AMS Subject Classification: 12J99, 46L99, 46S10 Key Words: p-adic Banach algebras, p-adic UHF algebras, p-adic spectral theorem, p-adic triangular UHF algebras 1. Introduction In an germinal 1960 paper [6] J. Glimm introduced the class of uniformly hyper- finite (UHF) C∗-algebras. In that paper Glimm showed that UHF algebras are classified up to ∗-isomorphisms by their supernatural numbers. Glimm’s classi- fication of UHF C∗-algebras inspired the author of the present paper to intro- duce in [2] a class of non-selfadjoint operator algebras that are currently called standard triangular UHF (TUHF) algebras. Briefly: a standard triangular UHF (TUHF) operator algebra is any unital Banach algebra that is isometrically iso- morphic to a Banach algebra inductive limit of the form T = lim(Tp ; σp p ); −→ n n m where (pn) is a sequence of positive integers such that pm | pn whenever m ≤ n. c 2012 Academic Publications, Ltd. Received: November 11, 2011 url: www.acadpubl.eu 20 R.L. Baker Here Tpn is the algebra of pn × pn upper triangular complex matrices and for m ≤ n, σpn pm : Tpm → Tpn is the mapping x ֒→ 1 ⊗ x = diag(x,...,x). The main result in [2] is that two standard triangular UHF operator algebras are iso- metrically isomorphic if, and only if they have the same supernatural number. This result can be extended in at least three different directions. One direction is to view standard triangular UHF operator algebras as special cases of Banach algebra inductive limits of upper triangular matrix algebras, and the the goal is to use purely Banach-algebraic methods to extend the main result in [2] to these inductive limits. Such an extension is presented in the paper [3], where it is proved that the supernatural number associated to an arbitrary triangular UHF (TUHF) Banach algebra is an isomorphism invariant of the algebra, pro- vided that the algebra satisfies certain “local dimensionality conditions.” The proof of main result of [3], although relying only on classical complex Banach algebra techniques, makes essential use of the classical spectral theorem (the Riesz functional calculus) for complex Banach algebras. A second direction in which the main result in [2] has been extended is to view standard TUHF algebras as triangular subalgebras of UHF C∗-algebras, such that the diagonal of the triangular subalgebra is a canonical masa in the ambient C∗-algebra. In this vein, the results in [2] have been substantially extended by J.R. Peters, Y.T. Poon and B.H. Wagner [11], and by S.C. Power [9], [10]. Finally, a third direction in which the principal results of [2] can be extended is to convert these results into a “pure piece” of p-adic functional analysis (to borrow a turn of phrase used by I. Kaplansky). For any prime number p, let Ωp be the p-adic counterpart of the complex numbers C (see [7], page 13). Replacing C by Ωp in the definition of complex (T)UHF Banach algebras, we obtain the definition of p-adic (T)UHF Banach algebras. The results in [2] for complex TUHF algebras can be duplicated for p-adic TUHF algebras, provided that a sufficiently gen- eral p-adic version of the Riesz functional calculus can be developed for p-adic Banach algebras ([3]). The purpose of the present paper it to prove a p-adic spectral theorem (Theorem 2.10) that is an extension of a classical p-adic spec- tral theorem of M. M. Vishik (see [7], page 149). We then use this extended p-adic spectral theorem to prove a p-adic Spectral Theorem for p-adic TUHF and UHF Banach algebras (Theorem 3.3). In the preprint [4] we use Theorem 3.3, Theorem 3.4, and Lemma 3.5 to transfer, mutatis mutandis, the results of [3] to p-adic TUHF Banach algebras. In the remainder of the present section of the paper we put forth the prelim- inary material on p-adic analysis and p-adic Banach algebras that is necessary for proving Theorem 2.10. Then in Section 2 we prove Theorem 2.10. Finally, in Section 3, we use Theorem 2.10 to prove Theorem 3.3. A CERTAIN p-ADICSPECTRALTHEOREM 21 Let p be a prime number, and let Q be the field of rational numbers. Let | · |p be the function defined on Q by a = pordpb−ordpa, |0| = 0. b p p Here ordp of a non-zero integer is the highest power of p dividing the integer. Then | · |p is a norm on Q. The field Qp is defined to be the completion of Q under the norm | · |p. Define Qp to be the algebraic closure of Qp (see [7]: p. 11). Unlike the case of the real numbers R, whose algebraic closure C is only a quadratic extension of R, the algebraic closure Qp of Qp has infinite degree over Qp. However, the norm | · |p on Qp can be extended to a norm | · |p on Qp. But it turns out that Qp is not complete under this extended norm. Thus, in order to do analysis, we must take a larger field than Qp. We denote the completion of Qp under the norm | · |p by Ωp, that is, ∧ Ωp =Qp, ∧ where means completion with respect to | · |p. The symbol “C” is usually used to denote Ωp (see [7], page 13 and [12], page 140). Definition 1.1. Let r ≥ 0 be an non-negative real number, and let a ∈ Ωp and let σ ⊆ Ωp. We have the following definitions. Da(r)= { x ∈ Ωp | |x − a|p ≤ r }; − Da(r )= { x ∈ Ωp | |x − a|p < r }; Dσ(r)= { x ∈ Ωp | dist(x,σ) ≤ r }; − Dσ(r )= { x ∈ Ωp | dist(x,σ) < r }. − − − If b ∈ Da(r), then Db(r) = Da(r), and if b ∈ Da(r ), then Db(r ) = Da(r ). − Thus, any point in a disc is its center. Hence Da(r), Da(r ) are both open and closed in the topological sense. It is conventional to take inf ∅ =+∞, hence for − r> 0 we have D∅(r )= ∅ and D∅(r)= ∅ (see [5], page 4). Lemma 1.2. Let ∅ 6= σ ⊆ Ωp be compact. Then for any s> 0 there exists 0 < r ∈ |Ωp|p and a1, ..., aN ∈ σ such that r<s and N Dσ(r)= Dai (r); i [=1 22 R.L. Baker N − σ ⊆ Dai (r ), i [=1 where the Dai (r) are disjoint. Proof. Since σ is compact, there exist a1, ..., aN ∈ σ such that N − σ ⊆ Dai (s ), (1) i [=1 − where the union is disjoint and for 1 ≤ i ≤ N, σ ∩ Dai (s ) 6= ∅. Now, for each − − 1 ≤ i ≤ N, Dai (s ) is closed, and hence σ ∩ Dai (s ) is compact. Therefore − there exists b1, ..., bN ∈ σ such that for 1 ≤ i ≤ N, bi ∈ σ ∩ Dai (s ) and − |bi − ai|p = sup{ |x − ai|p : x ∈ σ ∩ Dai (s ) }. − Because each bi ∈ Dai (s ), we have, for 1 ≤ i ≤ N, s > |bi − ai|p. Now select any r ∈ |Ωp|p such that r> 0 and s>r> |b1 − a1|p,..., |bN − aN |p. We claim that N − σ ⊆ Dai (r ), (2) i [=1 where Da1 (r),...,DaN (r) are disjoint. To prove the claim, let a ∈ σ; then by − − (1), a ∈ Dai (s ) for some 1 ≤ i ≤ N, and hence a ∈ σ ∩ Dai (s ), which implies that − |a − ai|p ≤ sup{ |x − ai|p : x ∈ σ ∩ Dai (s ) } = |bi − ai|p < r. − This shows that a ∈ Dai (r ), which proves (2). Now let 1 ≤ i, j ≤ N, with i 6= j. Suppose that Dai (r) ∩ Daj (r) 6= ∅, and let a ∈ Dai (r) ∩ Daj (r). Then we have |ai − aj|p ≤ max{ |ai − a|p, |a − aj|p }≤ r <s, − − hence ai ∈ Dai (s ) ∩ Daj (s ) = ∅. This contradiction proves the claim. We now claim that N Dσ(r)= Dai (r). (3) i [=1 A CERTAIN p-ADICSPECTRALTHEOREM 23 N It is clear that Dai (r) ⊆ Dσ(r). Let x ∈ Dσ(r), then because σ is compact, i=1 there exists a ∈Sσ such that r ≥ dist(x,σ)= |x − a|p. Hence, x ∈ Da(r). Now, − a ∈ σ, so by (2), there exists 1 ≤ j ≤ N such that a ∈ Daj (r ) ⊆ Daj (r). N Therefore, Da(r) = Daj (r), which shows that x ∈ Dai (r). This proves i=1 (3). S Lemma 1.3. Let ∅ 6= σ ⊆ Ωp. Let r> 0 and let b1, . , bM be in Ωp, with M − σ ⊆ Dbi (r ), (1) i [=1 where the Dbi (r) are disjoint. Then there exist a1, . , aN in σ and ∅ 6= I ⊆ { 1,...,M } such that the Dai (r) are disjoint and N − − − Dσ(r )= Dai (r )= Dbi (r ); (2) i i I [=1 [∈ N Dσ(r)= Dai (r)= Dbi (r). i i I [=1 [∈ Proof. Let ∅ 6= I ⊆ { 1,...,M } be the subset of { 1,...,M } consisting of − those 1 ≤ i ≤ M for which σ ∩ Dbi (r ) 6= ∅.
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